Prove that:

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Solution

Consider the left hand side (LHS),

LHS= ( sin7x+sin5x )+( sin9x+sin3x ) ( cos7x+cos5x )+( cos9x+cos3x ) = [ 2sin( 7x+5x 2 )cos( 7x−5x 2 ) ]+[ 2sin( 9x+3x 2 )cos( 9x−3x 2 ) ] [ 2cos( 7x+5x 2 )cos( 7x−5x 2 ) ]+[ 2cos( 9x+3x 2 )cos( 9x−3x 2 ) ] = ( 2sin6xcosx )+( 2sin6xcos3x ) ( 2cos6xcosx )+( 2cos6xcosx ) = 2sin6x( cosx+cos3x ) 2cos6x( cosx+cos3x )

Further simplify,

=tan6x

Right hand side (RHS),

RHS=tan6x

Since, LHS=RHS

Hence, the expression has been proved.

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